What is Valentine's role in the play?
He hardly has one. He's one of Orsino's servants, and he first appears having come back from Olivia's, to tell Orsino that she's not interested in him:
So please my lord, I might not be admitted;
But from her handmaid do return this answer:
The element itself, till seven years' heat,
Shall not behold her face at ample view;
But, like a cloistress, she will veiled walk
And water once a day her chamber round
With eye-offending brine: all this to season
A brother's dead love, which she would keep fresh
And lasting in her sad remembrance.
His name points to the patron saint of love: Valentine. And Valentine's job has clearly been to go to Olivia's and plead Orsino's love - and try and get him to agree to marry him and return his love. But it isn't working: she won't even now let him in.
So Cesario (really Viola) gets almost immediately given the job. And you might sense a little professional jealousy in what he first says to Viola in Act 1, Scene 4:
If the duke continue these favours towards you,
Cesario, you are like to be much advanced: he hath known you but three days, and already you are no stranger.
Viola clearly senses something slightly aggressive in him: is the Duke inconstant, in his favours, she asks? Valentine replies
No, believe me.
And that's all we see of him. Because, after that, it is Viola who is sent to Olivia: Orsino believes that Viola's youth will do more good than a 'nuncio of more grave aspect'. Perhaps Valentine is just too old.
But that is his role in the play - it's not large! And I've quoted, in this post, every line he speaks in the play!
Hope it helps!